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Full text of "The personal equation"

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The Popular science monthly. 

New York,Popular Science Pub. Co., etc. 
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V. 6 (1874-1875): http://www.biodiversitylibrary.org/item/17694 

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THE 



POPULAR SCIENCE 



MONTHLY. 



PEBRTTARY, 1875, 



THE PEESONAL EQUATIOIS". 

By Pbof. EDWARD S. HOLDEN, 

OF THE UNITED STATES NAVAL OBSERVATORY, WASHINGTON, D. C. 

^N" an attemj)! to explain clearly some of the phenomena wbich 
have led to the consideration of what astronomers call the "per- 
sonal equal ion," it will, perhaps, be most advantageous to consider 




the subject somewhat in an historical manner. In this way we shall, 
it is true, lose something in directness, but it will assist in gaining a 



definite conception of the whole subject if we consider it in the order 
in which astronomers have been forced to do. 

To make the meaning of the term plain, it will be necessary to 
premise a brief account of the methods of observation with astronomi- 
cal instruments, and of some of the refinements which have gradually 
been found necessary in these methods, 

Nearly every astronomical observation has for an object to fix the 



relative position of two bodies at a given time. If, then, a second 
observation of a similar kind is made, these two, taken together, will 
suffice to give some idea of the apparent relative motion of one body, 
referred to the other. If, for example, the design is to determine the 
orbit of a new comet, the mode of proceeding is, or might be, some- 
thing as follows : Some star, whose place is known (or whose place is 
subsequently determined), is chosen in the vicinity of the comet, and 
the distance of the comet from this star is measured. This may be 
done in several ways, by a sextant, with which we can measure this 
distance directly, or, more usually, by one of the fixed instruments of 
an observatory, with which we can determine two things: 1. The 
distance of the comet east or west of the star; and, 2. Its distance 
north or south of it. The distance north or south is usually deter- 
mined by a direct measure of the celestial arc included between the 
respective parallels on which the star and comet are at a given time ; 
while the distance east or west is usually measured by the interval 
of time required for the earth's rotation to carry a body from the 

TOL. VI. — 25 



386 



THE POPULAR SCIENCE MONTHLY. 



meridian of the star to that of the comet. To make this measure, it 
is customary to fix in the focus of the telescoj^e some uneven number 
of fine filaments of spider's-web at (say) equal distances apart, and to 
allow the telescope to remain fixed while the diurnal rotation of the 
earth carries the body first to be observed iuto the field of the tele- 
scope and slowly across this. As it crosses each of the threads, the 
time at which it is exactly on the thread is noted. Now^ when the 
second body enters the field of the telescope (which is supposed to 
remain fixed in its former position) the times of its passage over the 
various threads are noted. 

The mean of the times for the first body gives the time at which 
this body was on the middle thread (these being at equal intervals), 
while the mean of the times for the second body gives the cori'cspond- 
ing time for the second body, and the difierence of these two times 
gives evidently the distance which one of them is, east or west, of the 
other, expressed in time. This may be easily reduced to degrees^ etc., 
by the rule that twenty-four hours is equal to 360 degrees. 

If it were possible for an astronomer to note the exact instant of 
the transit of a star over a thread, it is plain that one thread would be 
sufficient; but, as all estimations of this time are, from the very nature 



of the case, but approximations, several threads are 



erted in order 



that the accidental errors of estimations may be eliminated, as far as 

The method of making these estimations will be better 
understood from the two following figures, 1 and 2. Fig. 1 represents 





! \ 



I 




ill \\\\\ Mill 



Fig. 1. — Transit-thbeads in Telescope. 



the reticle of a transit-instrument as it would be viewed by an observer, 
where twenty-five threads are placed arranged in groups or tallies of 
five. The star may enter on the left hand in the figure, and may be 
supposed to cross each of these wires, the time of its transit over each 



THE PERSONAL EQUATION. 387 

of tbem, or over a suflScient niiinber, being noted. The method of 
noting this time may be best understood by referring to Fig. 2. 

Supi)ose that the line in the middle of the figure is one of the 
transit-threads, and that the star is jDassing from the right hand of the 
figure toward the left : if it is on this wire at an exact second by the 
clock (which is always near the observer, beating seconds audibly), 
this second must be written down as the time of the transit over this 
thread. As a rule, however, the transit cannot occur on the exact beat 
of the clock, but at the seventeenth second (for example) the star will 
be on the right of the wire, say at a; while, at the eighteenth second, it 
will have passed this wire and may be at h. If the distance of a from 
the wire is six-tenths of the distance ah^ then the time of transit is to 



be recorded as — hours — minutes (to be taken from the clock-face), 
and seventeen and six-tenths seconds^ and in this way the transit 
over each wire is observed. This is the method of " eye-and-ear '' ob- 
servation, the basis of such work as we have described, and it is so 
called from the part which both the eye and the ear play in the appre- 
ciation of intervals of time. The ear catches the beat of the clock, 
the eye fixes the place of the star at a ; at the next beat of the clock 
the eye fixes the star at 5, and subdivides the space a h into tenths, at 
the same time appreciating the ratio which the distance from the 






o a 



Fig. 2.— Passage op Stab across the Thread. 



thread to a bears to the distance ah. This is recorded as above. 
Now, if the action of the eye and the ear and the coordinating action 
of the brain (which must associate some spot in the field of view with 
some second) were all instantaneous in their action, the phenomenon 
of personal equation would not exist. As a matter of fact, when the 
clock beats and the star is really at a, the mind refers it to some point 
farther on in the field as a'/ and when the clock again beats, the star, 
which truly is at J, is by the mind referred to a point b\ The dis- 
tance a h is the same as a^ V ; but the distance from the thread to a is 
greater than the distance from the thread to a\ Hence, instead of 
recording the time of transit as 17^6, an observer, whose habit is cor- 
rectly represented by the figure, might record this time as 17\4, and 
the correction + 0^2 would be required to be applied to his times of 
transit to reduce them to the exact truth: -f 0^2 is then his absolute 



personal correction. But, in general, we have no means of determin- 



ing where a and &, in our field of view, are, and hence the knowledge 
of the ahsolute personal equation has to be gained by some special de- 



\ 



388 THE POPULAR SCIENCE MONTHLY. 

vices, to be hereafter spolcen of. A little consideration will sliow, how- 
ever, that, although every transit observed by our astronomer is too 
early by 0\2, yet, in ordinary cases, this correction is of no account, 
provided only that it is constant. If he observes the star too early by 
0®.2, and the comet also too early by that amoimt, the difference in the 
times will be absolutely correct. But suppose one observer to note 
the transit of the star, and another that of the comet : each may 
have a peculiar habit, so that where one would note 0^2 top early, 
another miglit note 0^3 too early, and the diftWence of their absolute 
personal equations, OM, it would be necessary to apply to the observa- 
tions of A to reduce them to homoGjeneousness with those of B. This 
difference of absolute i3ersonal equations is relative personal equation, 
which, when once truly known, enables us to reduce the observations 
of one skillful astronomer to what they would have been had another 
made tliem. * 

We say " skillful,'' because it is only among skillful observers that 
the phenomenon in question is truly found. In astronomical obser- 
vations the senses are trained to a fine delicacy, and old observers 
acquire a constancy of habit which gives to their work a homo- 
geneousness that is wanting in that of younger men. 

We have given a brief account of the early method of estimating 
the time of a star's transit across a spider-line in the field of the 
telescope by the method of eye and ear; there is yet another method 
now in common use, which it is necessary to understand before we 



pass to the consideration of the means of determining personal 
equation. 

This second method is the American or chronographic method ; 
this consists, in the present practice, in the use of a sheet of paper 
wound about and fastened to an horizontal cylindrical barrel, which is 
caused to revolve by machinery once in one minute of time. A pen 
of glass which will make a continuous line is allowed to rest on the 
paper, and to this pen a continuous motion of translation in the direc- 
tion of the length of the cylinder is given. Now, if the pen is allowed 





to mark, it is evident that it will trace on the paper an en 
spiral line. An electric current is caused to run through the observing 
clock, through the pen, and through a key which is held in the ob 
server's hand. 

A simple device enables the clock every second to give a slight 
lateral motion to the pen, which lasts about a thirtieth of a second. 
Thus every second is automatically marked by the clock on the chro- 
nograph-paper. The observer also has the power to make a signal 
(easily distinguished from the clock-signal by its different length), 
which is likewise permanently registered on the sheet. In this way, 
after the chronograph is in motion, the observer has mei'ely to notice 
the instant at which the star is on the thread, and to press the key 
at that moment. At any subsequent time he must mark some hour. 



THE PERSONAL EQUATION. 



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390 THE 



M ONTHL 



minute, and second, taken from the clock on the sheet at its appropriate 
place, and the translation of the spaces on the sheet into times may be 
done at leisure. This will be plainer if we examine Fig. 3, which is a 
fac-simile of a portion of a chronographic record. 

The marks of the clock giving regular signals of seconds are easily- 
distinguished ; the rattles were made by the observer to attract atten- 
tion to an observation to follow : and the signals of the observer are 
seen at the times of the transits of various stars. By applying a grad- 
uated ruler to the sheet when it is unrolled, the exact times of transit 
can be determined to within a hundredth part of a second, provided 
we have some hour, minute, and second marked on the sheet as an 
origin of time. 

It is quite plain that the senses of the observer are not strained to 
so great a degree in this method as in the method of eye and ear ; the 
eye has but one thing to do, the ear is not occupied, and the hand has 
only to press the key at the proper time. 

In this method, we see that the origin of relative personal equation 
is again in the different times required for different observers to co- 
ordinate the position of the star in the field and the position of the 

wire. 

True personal equation, considered physiologically, must arise from 
the personal differences between observers wiien they note the smne 
phenomenon. With the chronograph it is the habit of most observers 
to tap the observing key at the moment at w^hich the star is actually 
on the wire. There are cases, however, where astronomers of some 
experience are accustomed to taj) the key so that the sound of the tap 
shall come to the ear at the time w^hen the star is on the wire. This 
seems an utterly wrong habit of observing, as it is really the record 
of an event which has not yet taken place which such an observer 
makes. Astronomically, the difference between such an observer and 
another observer may be treated as a case of personal equation, pro- 
vided the habit described above remains constant, which it is proba- 
bly less likely to do than the ordinary one. 

The first case of personal equation on record aj)pears in the " Ob- 
servations " of the Rev, Nevil Maskelyne, Astronomer Royal for Eng- 
land ("Observations" for 1196, vol. iii., p. 339), We there find the 
wing note: "I think it necessary to mention that my assistant, 
Mr. David Kinnebrook, who had observed the transits of the stars 
and planets very well in agreement with me all the year 1794, and for 
a great part of the present year, began from the beginning of August 
last to set them down half a second of time later than he should do 
according to my observations ; and, in January of the succeeding 
year, 1796, he increased his error to eight-tenths of a second. As he 
had unfortunately continued a considerable time in this error before 
I noticed it, and did not seem to me likely ever to get over it and 
return to a right method of observing, therefore, though witli reluc- 




THE PERSONAL EQUATION. 391 

tance, as he was a diligent and useful assistant to me in otlier respects, 
I parted with him." 

But time has its revenges, and Kinnebrook's observations are now 
used as well as Maskelyne's {see " Annales de PObservatoire de Paris j 
Memoires," iii., p. 307), and they are probably about as free from acci- 
dental errors as his. 

In 1822 Bessel examined this subject, and we find in the Konigs- 
berg observations of that year an account of quite extended experi- 
ments on personal equation. 

Bessel, after quoting from Maskelyne's own report (see extract 
above), considers the subject at some length. He calls attention to 
the fact that the accidental errors in an eye-and-ear observation cer- 
tainly do not exceed two-tenths of a second, and that a careful con- 
sideration of the observations of Maskelyne and his assistant shows 
that there may be an " involuntary constant difference " between the 
estimations of various observers which far surpasses the limits of pos- 
sible accidental error. 

In 1819 Bessel made a visit to the Seeberg Observatory, where he 
observed, on two nights, transits with Von Lindenau and Encke. 
These observations showed no personal equation between these three 
celebrated astronomers. In 1820 Dr. Walbeck and Bessel made sev- 
eral sets of observations at Konigsberg, for the purpose of determining 
their relative personal equation, and the results of their work are 
given below : 

8. 

1820, December 16th and l'7th, Walbeck later than Bessel 1.045 



<£ T7+V» or.^ iQfV. <« « « (i 



iTth and 19 th, 



* «• * * •« • • * Vti/Otl 



19th and 20th, " " « " l.oio 

1.025 



<i OnfV» nr^A OOrl U « « C( 



20th and 22d, 



Bessel says that this great difference was evident from the second 
day, and that no pains was spared by either of them to observe care- 
fully ; and that at the end of the series each was confident that it would 
have been impossible for him to observe differently, by so much even 
as a teuth of a second. Here, then, was an enormous difference— one 
almost incredible. To test the reality of the phenomenon, Bessel com- 
pared with Argelander, and found that Argelander was later than he 
by 1^223. 

Bessel remarks that neither Walbeck nor Argelander had observed 
as much as he had with the transit-instrument, and he therefore used 
all opportunities for comparing his work with that of Struve, of 
Dorpat. He found that in 1814 Struve was later than himself by 
0\044; in 1821, by 0^799 ; in 1823, by 1^021. Bessel now determined 
to arrive at some conclusion by studying this phenomenon under dif- 
ferent aspects. 

To this end Argelander and himself noted the times of 78 disap- 



392 THE POPULAR SCIENCE MONTHLY. 



pearances or reappearances of a material object, and he fouiicl that 
Argelander was later than himself by 0',222. Again, in the^ obser- 
vation of the occultations of stars (an instantaneous phenomenon), 
Argelander was slower than Bessel by 0'.281. Here was some light: 
for it was now evident that not only had each astronomer a diflerent 
habit of estimating time, but that this habit was only constant so long 
as the same phenomenon was observed ; that a j)ersonal equation for 
transit observations would not serve for observations of occultations. 
Bessel next investigated the question whether there was any dif- 
ference in his own absolute personal equation in observations with 



a clock beating whole seconds, or with a chronometer beating half- 




s ; he found that he observed 0'.494 later when tlie clock beat 
half-seconds than when it beat whole seconds, while Argelander and 
Struve did not change their habits in this regard. 

Bessel's whole investigation is very complete, especially when we 
consider that it was the first published research on a subject which 
had escaped attention until his time. The principal points established 
were : 

1. A personal equation subsists in general between two observers. 

2. For limited periods of time this equation is probably constant 
between two observers for the same class of work. 

3. The absolute personal equation of any one observer varies with 
the class of observation; i, e., from transit observations to sudden 
phenomena like occultations. 

4. The rapidity with which the star (in transit observations) trav- 
ersed the field of the telescope had no influence on Bessel's personal 
equation. 

Bessel does not seem to have supposed that there would be any 
different personal equation for stars and for the moon. This we now 
know to have been erroneous, and we shall see that the apparent velo- 
city with which a star moves through the field of the telescope is also 
held by some observers to have an influence on the magnitude of their 
personal equation. 

All of the preceding results referred simply to the personal equa- 
tion between observers who were using the eye-and-ear method. As 
soon as the chronographic method of registering transits was intro- 
duced, it was seen that the personal equation became smaller. This 
is undoubtedly due to the smaller amount of work which the brain has 
to perform ; the phenomena to be appreciated are, in this latter case, 
far more simple than in the former, and the effect of this is shown in 
the amount of personal difference. 

We must now give a brief account of the ordinary methods for 
determining the amount of the relative personal equations of various 
observers, in order that we may proceed to the determination of the 
absolute equation, which is of great interest physiologically and psy- 
chologically, although not of capital importance to astronomy. As 



THE PERS02{AL EQUATION. 




we have seen, to reduce the observations of A to what they would 
have been if B had made them, it is simply necessary to know how 
much later B is in the habit of observing than A, and to apjjly this as 
a constant correction to A's work. 

This may be done in practice by A and B observing the same star 
in the same telescope; A over tlie first ten wires [see Fig. 1), and B 
over the second ten. 

A knowledge of the distances of the various wires from the middle 
wire enables ns to compare A's worlf with B's, and A— B is the rela- 
tive personal equation. 

There is, however, a strong objection to this process: if personal 
equation is any thing, it is the difference between established habits y 
and, if A observes over ten wires, and then hastily rises to allow B to 
take his place at the instrument, both A's habits and B's are broken 
in upon, and the resulting personal equation is likely to be affected by 
this fact. In general, the way adopted is to allow A to observe sev- 
eral stars leisurely, and from them to determine the error of the clock j 
B does the same, and from his observations also a clock-error is found; 
the difference of these clock-errors, reduced to the same epoch, gives 
the relative equation of A and B. 

Now if, instead of A registering his own observations on the chrono- 
graph (for example), we could have the star register its own transit, 
then B's observation, compared with this, would give at once an abso- 
lute equation. We cannot use the real star for this purpose ; but sev- 
eral attempts have been made to construct an apparatus which should 
register the transit of an artificial star, which star could, at the same 
time, be observed. The principle of all of these machines is, in gen- 
eral, the same, and we will merely give a brief account of one which 
is now under trial by the Coast Survey. 

The artificial star is produced by lamp-light falling upon a small 
hole in a blackened plate ; this plate is given a motion laterally, and 
the small point of light passes from one side to the other of a plate of 
ground glass, upon which lines are ruled to represent the spider-lines 
of the reticle. As the artificial star passes each wire, an electric signal 
is recorded on the chronograph, and the observer can also record his 
signal; and thus on the same chronograi^h-sheet many observations 
of absolute personal equation can be permanently recorded. Any 
velocity can be given to the star, so that it may pass through the 

of view as slowly as the pole-star, or as rapidly as a star at th 
equator.* 

^ The chief objection to this apparatus is, that there is a constant error in its indica- 
tions ; i. e., it can never be adjusted so as to give its signal at the exact moment of tran- 
sit, but It is always too soon or too late. This is sought to be eliminated by allowing the 
artificial star to travel first from right to left, and then from left to right, and using the 
mean of the two determinations. It is still a question whether the observer's habit is the 
same no matter which way the star is moving. 




e 



^ 



394 THE POPULAR SCIENCE MONTHLY. 

An apparatus similar to this ^\#s invented and used by Wolf, of the 
Paris Observatory, and we owe to him much the fullest account of 
personal equation which we have. We cannot do better than to give 
a brief abstract of his memoir ("Memoires de I'Observatoir e de 
Paris,'' tome viii., p. 153), as the results obtained by the American de- 
vice have not been made public. 

His first experiences showed him that his absolute personal equa- 
tion, when he used the chronographic method of recording, was ex- 
tremely small (from three to foicr hundredths of one second) ; and, 
although this was an interesting fact, yet the very smallness of this 
equation showed that it was hopeless to attempt to discover the laws 
of variatio7i of so minute a quantity. 

These laws would be masked by the accidental errors : so that all 
the observations of M. Wolf have been by eye and ear. It should be 
stated that M. Wolf is an observer of experience. In his own experi- 
ments he proposed to himself to determine the effect on his equation — 

{a.) Of the position of the observer (sitting or standing, etc.). 

{p.) Of the magnifying power of the telescope. 

(c.) Of the direction of motion of the star (L e., whether from right 

to left, or the reverse). 

{(L) Of the brightness of the star. 

His personal equation he found was, at first, about +0^3 j and in 
a short time this fell to +0M; this was undoubtedly due to the fact 
that the observer felt in what direction his observations had to be 
modified, in order to bring them nearer to the truth, and that he un- 
consciously so modified them. This, however, did not continue with- 
out limit ; his personal equation remained, for all the time he observed, 
at this lower limit, and this fact gave him the first clew to the physio- 
logical explanation of the phenomenon. 

M. Wolf finds that the brilliancy of the star has no sensible effect 
on personal equation, a conclusion identical with that derived by Mr. 
Dunkin, of the Royal Observatory at Greenwicli ("Monthly Notices, 
Royal Astronomical Society," vol. xxiv., p. 158), 

With regard to the influence of the direction of motion of the arti- 
ficial star, M. Wolf finds in his own case a mean constant difference 
of 0'.04 obtained from over 400 transits : this he subsequently explains 
by the fact that, if his right eye be fixed on two dots equidistant from a 
line drawn on a sheet of paper, one of these dots always ajDpears nearer 
to the line than the otlier by a small quantity. This, of course, is a 
defect in the symmetry of the eye, and it is quite a common defect, 
which probably many of the readers of The Popflae Science Month- 
ly have, perhaps without knowing it. 

The influence of the apparent velocity of the star Bessel states to 
have been nothing in his own case, provided the star was situated 
more than 20° from the pole. Wolf's experiments do not agree with 
this, and he confirms the researches of Dr. Pape and of Dunkin. 



THE PERSONAL EQUATION. 395 

Pape finds {Astronomische N'achrichten^ vol. xliv., p. 179) that the 
error of a transit observation is composed of two parts : one is con- 
stant, and the other depends on the polar distance of the star. Dunkin 
likewise considers the probable error of a transit observation as depend- 
ing upon the polar distance of the star, and Wolf's experiments cor- 
roborate these results, and show that his own personal equation became 
larger as the velocity of the star increased. It is evident that this 
rule must be held true only within limits, and probably these limits 
are not very far apart. Wolf further made experiments to determine 
whether the position of the observer affected his personal equation, 
and he concluded that, for his own case, there was no effect due to 
this cause. It is probable that most astronomers would differ with 
Wolf in this respect : observers of double stars, especially, have no- 
ticed a constant influence in their measures due to the position of the 




After having recited the results of his experiments, M. Wolf comes 
to the consideration of the really important question, " What is tbe 
origin of the phenomenon known as personal equation?'' Before he 
discusses this, he considers the remarkable personal differences be- 
tween Bessel and other astronomers which we have noticed, showing 
that this is undoubtedly the largest personal equation on record, and 
expressing his opinion that it was really due to an erroneous counting 
of the whole seconds, and that the fractional part of his enormous per- 
sonal equation with Argelauder (1\223) was alone a case of true physi- 
ological personal difference. Let us recall the fact that Bessel and 
Argelander differed in observations of sudden phenomena only by 
0\222, or 0'.281 ; and again, that Bessel observed transits with a chro- 
nometer beating half-seconds so much as 0*.494 (nearly a whole beat) 
later than with a clock beating seconds ; and it seems impossible to 
avoid Wolf's conclusion that Bessel counted his seconds differently 
from other observers. The only thing which militates against this 
theory is, that Bessel must have examined tliis question of enumera- 
tion himself; and again, that, in two nights' observation with Von Lin- 
denau and Encke, he found no signs of personal equation. Encke, 
however, in speaking of this large personal equation of Bessel's, says 
that there is no doubt that he had a different method of countins the 
strokes of the clock from other observers. M. Wolf, too, mentions 
tbe case of an assistant at the Paris Observatory, whose transit ob- 
servations were earlier by one second than those observed by his fel- 
low-assistants (Bessel's habit), but, in this case, a few experiments on 
artificial transits sufliced to show him that his habit was wrong, and 
led him to chancje it. 

The opinion of most astronomers has been, that personal equation 
is not purely a physiological phenomenon, but likewise a psychological. 
The time required for the sound of the clock to reach the observer's 
brain, and the time required for the light to pass from the image of 



^ 




THE POPULAR SCIENCE MONTHLY. 



the star, so as to excite the nerves of vision, are both very small : it is 
the coordinating j)ower of the brain that works slo*wly — and absolute 
jDersonal equation is largely the measure of the time required for the 
brain to superpose two difterent sensations^ to coordinate impressions 
derived from different sets of nerves. 

This view M. Wolf combats, and maintains, on the contrary, that 
the phenomenon in question is purely physiological, and arises from 
the duration of the luminous impression of the image of the star on 
the retina. To prove this, he has applied his apparatus to the observ- 
ing of transits in which the seconds of the clock were not marked by 
audible beats, but by flashes of light appearing in the field of the 
telescope. 

In this case, and also in the case where the seconds of the clock 
were not heard, but were marked by light taps on his hand, his equa- 
tion remained almost constant {see table) : 

The second marked by sound, • £=: + CMC 80 observations. 



a c( 



** '' sight, : e= + 0.08 80 



(C 



The second marked by sound, : e = + OMl 80 " 

*' *' " *' feeling, :c= + 0,ll 80 " 

This table seems to bear out M. Wolf's view ; but, in this connec- 
tion, it will be interesting to refer to a ])aper by Mr. T. C. Mendenhall, 
of Columbus, which aj)peared in the American Journal of Science^ voL 
ii., p. 157. This gentleman says : '^ An attempt was made to determine 
the relative rapidity with which responses are made to impressions 
made upon the different senses. . . . Time is measured on a register 
similar to the astronomical chronograph, in which I have been able to 
move a slip of paper with great regularity at the rate of about one 
and one-half inch per second, the seconds being registered upon the 
slip by a seconds pendulum according to the electric method. The 
person on whom the experiment is being made is seated at a table, 
having his hand on a key ; by pressing this, the time of the action is 
registered on the paper. I made an apparatus, by means of which the 
circuit is completed for an instant the moment that there appears at 
a circular opening, about three-fourths of an inch in diameter, a card, 
red or white, as I choose, which completely fills the opening. The 
subject is instructed to watch this opening, and to press the key im- 
mediately on seeing the c^rd. The actual appearance of the card and 
his closing the circuit in response are marked on the slip of paper by 
two dots about one-fifth of an inch apart (two-fifteenths of a second). 
This is a measure of the time occupied by the somewhat complex 
operation of his perceiving the object, and acting in response to that 
perception. I introduce the exercise of judgment by giving him two 
keys, one for each hand, and by instructing him that, when a white 
card appears, he is to close with his right hand, and when a red card 
appears, with his left. . . , According to the same general plan, I 
made trials concerning the sense of hearing, I arranged tliat, by 



TEE PERSONAL EQUATION. 2>97 




pressing upon the key, unseen by tlie subject, I could at the same time 
close the circuit and produce a clear and distinct sound, upon hearing 
which he made a response registered as before. I connected my ap- 
paratus with the key-board of a piano-forte in such a way that I was 
able to introduce an exercise of judgment in the comparison of two 
tones differing in pitch much or little, as I chose, . . . With different 
persons, as many as 2,000 individual trials have been made, and the 
errors of experiment eliminated as far as possible by averages. . . . 
As was anticipated, different individuals furnished, in some cases, stri- 
kingly different results, but, in general, they all followed the order 
given in the table : '' 

CASE OF A. G. F. Time in seconds. 
Eesponse to appearance of a white card 0.292 

'* " . " " an electric spark (in the dark). , 203 

<i it QAIlTlfl 1^8 

" " touch on the forehead • . 

" " " " " hand...... 117 

'* when required to decide between white and red 443 

" " " " " . " tones C and E 335 

" " " " " " C and C above (octave) 42S 

One cannot but be struck with the additional time required when 
tlie phenomenon to be observed becomes even slightly more complex. 
This is evidently not entirely a physiological effect, but is truly j3sy- 
chological in part. Just what bearing this has on the question of the 
cause of personal equation it would be difficult to say : at the same 
time we must admit that the slightest additional exercise of judgment 
requires additional time. This is forcibly shown by the smallness of 
chronograpbic personal equation as compared to eye-and-ear-equation. 

Let us now consider personal equation in things other than the 
estimation of time. We stated that the distance of one star, north or 
south, of another, was usually measured directly ; i. e., by graduated 
circles for large distances, and with micrometers for small ones. Prof. 
Coffin, now Superintendent of the American Ephemeris, has shown 
that in his own case, and in the case of two other observers, at the 
United States Naval Observatory of Washington, a marked personal 
difference appears in the observations of a Lyr^e, and one or two other 
stars which pass near the zenith of Washington, depending on the 
direction in which the ohserver faced^ whether north or south. It is 
plain that a star near the zenith may be observed as a south star or as a 
north star, and it appears that each position gives a different polar dis- 
tance to the star: the difference of polar distance is small but constant. 

In reading microscopes, and, in short, in performing any operation 
where the senses are strained to appreciate small differences of time, 
space, or position, and particularly where the judgment has to be ex- 
ercised, personal differences are present. In general, these are con- 
stant with the same observer, and in astronomy thoy are usually 
eliminated in the determination of the zeros. For example, if an ob- 



% 




398 THE POPULAR SCIENCE MONTHLY. 

server reads the microscopes of a Transit Circle habitually too large, 
when he is determining the zenith-distance of a star, it is likewise 
his habit to read them too large when determining the position of the 
zenith-point from which zenith-distances are counted ; and the re- 
sulting quantity is likely to be free from all but accidental errors. 

Occasionally there arise cases where these differences (in the same 
observer) are not eliminated, but multiplied. 

In the measurement of a base-line, for example, the various rods 
are brought into contact under a microscoj^e : if an observer judges 
these rods to be in contact when they are not, it is evident that his 
error, originally small, will augment with the number of contacts, and 
it may become serious. 

In the comparison of the national standards of length, undertaken 
by the English Ordnance Survey, an annoying case of personal differ- 
ence was found. 

These comparisons were made by bringing a movable cross of spi- 
der-lines to bisect one of the lines engraved on the various bars, and 
it was found that Captain Clarke, R. E., and Quartermaster Steel, 
R. E., who made the greater number of comparisons, differed in their 
estimation of a bisection by a constant amount which was annoyingly 
large : so that " the probable error of the final results is nearly double 
what might be expected from errors of observations only." This error 
cannot be eliminated, and it still remains in the published results. 

We must constantly bear in mind that the quantities of which we 
have all along been speaking are extremely small, and that in fact 



they are masked by accidental errors for inexperienced observers m 
most cases. Still they exist, and they are among the most curious of 
phenomena : their careful study would well repay physiologists* 

We can never be sure we have eliminated them so long as the hu- 
man mind or body is a part of the machine by means of which we 
are comparing or registering events ; and, just so long as mind or body 
is employed, we can be sure that personal differences will not only 
exist, but that they will vary from day to day. We must use for 
eliminating personality those values which are the best attainable, 
and assume these values to be constant over extended periods of time 
— weeks or months. In astronomy of precision, however, we have 
other errors to fear much more variable than personal equation, and 
it is to the elimination of these that attention should be directed. In 
other branches of research less exact in method, personality becomes 
of more importance, and an attentive consideration of its effects may 
be well worth while undertaking.^ 



^ The writer has recently had occasion to examine drawings of the same nebula by 
different observers, with telescopes which are quite similar, and the enormous dififerences 
Avhich exist in the representations show personal diiFerences of the most marlvcd kind, 
for nothing is more certain than that all the changes shown by the drawings have not 
taken place.